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Theorem imasf1oxmet 24543

Description: The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
imasf1oxmet.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasf1oxmet.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasf1oxmet.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasf1oxmet.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasf1oxmet.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
imasf1oxmet.d	⊢ 𝐷 = (dist‘𝑈)
imasf1oxmet.m	⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))

**Assertion**
Ref	Expression
imasf1oxmet	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Proof of Theorem imasf1oxmet Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1oxmet.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasf1oxmet.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasf1oxmet.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		f1ofo 6828	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
6		imasf1oxmet.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7		eqid 2762	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
8		imasf1oxmet.d	. . . 4 ⊢ 𝐷 = (dist‘𝑈)
9	1, 2, 5, 6, 7, 8	imasdsfn 17574	. . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
10	1	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “_s 𝑅))
11	2	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
12	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵)
13	6	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍)
14		imasf1oxmet.e	. . . . . . . 8 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
15		imasf1oxmet.m	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
16	15	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉))
17		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
18		simprr 784	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
19	10, 11, 12, 13, 14, 8, 16, 17, 18	imasdsf1o 24542	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
20		xmetcl 24499	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ^*)
21	20	3expb 1137	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
22	15, 21	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
23	19, 22	eqeltrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
24	23	ralrimivva 3207	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
25		f1ofn 6821	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
26	3, 25	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
27		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏)))
28	27	eleq1d 2847	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
29	28	ralrn 7083	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
30	26, 29	syl 18	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
31		forn 6795	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
32	5, 31	syl 18	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
33	32	raleqdv 3322	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
34	30, 33	bitr3d 284	. . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
35	34	ralbidv 3187	. . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
36	24, 35	mpbid 235	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*)
37		oveq1 7419	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦))
38	37	eleq1d 2847	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
39	38	ralbidv 3187	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
40	39	ralrn 7083	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
41	26, 40	syl 18	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
42	32	raleqdv 3322	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
43	41, 42	bitr3d 284	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
44	36, 43	mpbid 235	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*)
45		ffnov 7538	. . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
46	9, 44, 45	sylanbrc 594	. 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ^*)
47		xmeteq0 24506	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
48	16, 17, 18, 47	syl3anc 1397	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
49	19	eqeq1d 2764	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝑎𝐸𝑏) = 0))
50		f1of1 6819	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–1-1→𝐵)
51	3, 50	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝐵)
52		f1fveq 7260	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
53	51, 52	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
54	48, 49, 53	3bitr4d 314	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
55	16	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐸 ∈ (∞Met‘𝑉))
56		simpr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
57	17	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑎 ∈ 𝑉)
58	18	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑏 ∈ 𝑉)
59		xmettri2 24508	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
60	55, 56, 57, 58, 59	syl13anc 1398	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
61	19	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
62	10	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑈 = (𝐹 “_s 𝑅))
63	11	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
64	12	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐹:𝑉–1-1-onto→𝐵)
65	13	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑅 ∈ 𝑍)
66	62, 63, 64, 65, 14, 8, 55, 56, 57	imasdsf1o 24542	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑎)) = (𝑐𝐸𝑎))
67	62, 63, 64, 65, 14, 8, 55, 56, 58	imasdsf1o 24542	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑏)) = (𝑐𝐸𝑏))
68	66, 67	oveq12d 7430	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) = ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
69	60, 61, 68	3brtr4d 5142	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
70	69	ralrimiva 3156	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
71		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑎)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑎)))
72		oveq1 7419	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑏)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))
73	71, 72	oveq12d 7430	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) = (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
74	73	breq2d 5120	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
75	74	ralrn 7083	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
76	26, 75	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
77	32	raleqdv 3322	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
78	76, 77	bitr3d 284	. . . . . . . 8 ⊢ (𝜑 → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
79	78	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
80	70, 79	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
81	54, 80	jca 520	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
82	81	ralrimivva 3207	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
83	27	eqeq1d 2764	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0))
84		eqeq2 2774	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
85	83, 84	bibi12d 348	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ↔ (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏))))
86		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → (𝑧𝐷𝑦) = (𝑧𝐷(𝐹‘𝑏)))
87	86	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
88	27, 87	breq12d 5121	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
89	88	ralbidv 3187	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
90	85, 89	anbi12d 643	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑏) → (((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
91	90	ralrn 7083	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
92	26, 91	syl 18	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
93	32	raleqdv 3322	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
94	92, 93	bitr3d 284	. . . . 5 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
95	94	ralbidv 3187	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
96	82, 95	mpbid 235	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
97	37	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷𝑦) = 0))
98		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥 = 𝑦 ↔ (𝐹‘𝑎) = 𝑦))
99	97, 98	bibi12d 348	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦)))
100		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑧𝐷𝑥) = (𝑧𝐷(𝐹‘𝑎)))
101	100	oveq1d 7427	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))
102	37, 101	breq12d 5121	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
103	102	ralbidv 3187	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
104	99, 103	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
105	104	ralbidv 3187	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
106	105	ralrn 7083	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
107	26, 106	syl 18	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
108	32	raleqdv 3322	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
109	107, 108	bitr3d 284	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
110	96, 109	mpbid 235	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
111	15	elfvexd 6917	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
112		focdmex 7951	. . . 4 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
113	111, 5, 112	sylc 66	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
114		isxmet 24492	. . 3 ⊢ (𝐵 ∈ V → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
115	113, 114	syl 18	. 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
116	46, 110, 115	mpbir2and 725	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 Vcvv 3454 class class class wbr 5108 × cxp 5658 ran crn 5661 ↾ cres 5662 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 –onto→wfo 6534 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7412 0cc0 11106 ℝ^*cxr 11248 ≤ cle 11250 +_𝑒 cxad 13141 Basecbs 17275 distcds 17325 “_s cimas 17564 ∞Metcxmet 21518

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-fz 13542 df-fzo 13690 df-seq 14045 df-hash 14374 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-0g 17500 df-gsum 17501 df-xrs 17562 df-imas 17568 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-xmet 21526

This theorem is used by: imasf1omet 24544 xpsxmet 24548 imasf1obl 24656 imasf1oxms 24657

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